Quantum-optimal-control-inspired ansatze for variational quantum algorithms

Alexandre Choquette,1, 2 Agustin Di Paolo,1 Panagiotis Kl. Barkoutsos,2

David Senechal,1 Ivano Tavernelli,2 and Alexandre Blais1, 3

1Institut quantique & Departement de physique, Universite de Sherbrooke, Sherbrooke J1K 2R1 QC, Canada2IBM Quantum, Zurich Research Laboratory, Saumerstrasse 4, 8803 Ruschlikon, Switzerland

3Canadian Institute for Advanced Research, Toronto, ON, Canada(Dated: August 5, 2020)

A central component of variational quantum algorithms (VQA) is the state-preparation circuit,also known as ansatz or variational form. This circuit is most commonly designed to respect thesymmetries of the problem Hamiltonian and, in this way, constrain the variational search to asubspace of interest. Here, we show that this approach is not always advantageous by introducingansatze that incorporate symmetry-breaking unitaries. This class of ansatze, that we call Quantum-Optimal-Control-inspired Ansatze (QOCA), is inspired by the theory of quantum optimal controland leads to an improved convergence of VQAs for some important problems. Indeed, we benchmarkQOCA against popular ansatze applied to the Fermi-Hubbard model at half-filling and show thatour variational circuits can approximate the ground state of this model with significantly higheraccuracy and for larger systems. We also show how QOCA can be used to find the ground stateof the water molecule and compare the performance of our ansatz against other common choicesused for chemistry problems. This work constitutes a first step towards the development of a moregeneral class of symmetry-breaking ansatze with applications to physics and chemistry problems.

The rise of noisy intermediate-scale quantum proces-sors [1, 2] requires us to find novel algorithms designedto attenuate the effects of noise. Variational quantum al-gorithms (VQA) are an example of such methods [3, 4].These algorithms make use of a (noisy) quantum com-puter and a classical co-processor to minimize a costfunction specified by a problem Hamiltonian Hprob. Thisminimization is achieved by preparing a state that ap-proximates the ground state of Hprob on the quantumcomputer using an iterative procedure driven by the clas-sical co-processor. Importantly, and thanks to the varia-tional nature of these algorithms, this approach has beenshown to potentially be resilient against noise, and well-suited to several applications including finance [5], puremathematics [6], machine learning [7, 8], optimizationproblems [9, 10], quantum chemistry and materials [11–15], as well as quantum optics [16].

In VQAs, the state preparation requires the parame-terization of a quantum circuit, referred to as the ansatzor variational form, that may or may not be structuredaround the problem. Recently, a considerable amount ofeffort has been invested in designing ansatze that pre-serve the symmetries of the problem Hamiltonian [17–22]. The goal of symmetry-preserving strategies is toconstrain the variational search to a small vector spaceof interest, which in principle improves the probabilityof convergence to the target state with fewer optimizeriterations.

In this work, we highlight shortcomings of this ap-proach. We then provide an ansatz that goes beyondsymmetry-preserving methods by introducing a set ofunitaries that break the symmetries of the problemHamiltonian. To achieve this, we borrow ideas fromthe theory of quantum optimal control, where fast andhigh-fidelity operations are achieved through the addi-tion of time-dependent symmetry-breaking terms to the

Hamiltonian. Focusing on fermionic systems, we incor-porate such terms in a time-evolution-like ansatz [23]to obtain the Quantum-Optimal-Control-inspired Ansatz(QOCA). We benchmark this approach against commonansatze found in the literature for the Fermi-Hubbardmodel and apply these ideas to the water molecule withminimal modifications. We find that in most cases, thismethod produces approximations of the target groundstate that are orders of magnitude more accurate. Tounderstand this improvement, we show evidence thatQOCA allows for an exploration in a slightly largerHilbert space.

The paper is organized as follows: in Sect. I, we beginby presenting known approaches to the construction ofthe ansatz and then formally introduce QOCA and quan-tum optimal control theory in Sect. II. We also elaborateon our strategy for the selection of symmetry-breakingterms in Sect. II and explain how these can be incor-porated into a variational ansatz for the Fermi-Hubbardmodel in Sect. III. Finally, we compare results obtainedwith the different approaches in Sect. IV.

I. VARIATIONAL ANSATZE

In the VQA framework, a quantum processor storesa quantum state |ψ(θ)〉 parametrized by a collection ofclassical variational parameters θ. This state is preparedfrom a known and easily prepared reference state, |ψ0〉,using a quantum circuit (the ansatz) U(θ) such that

|ψ(θ)〉 = U(θ)|ψ0〉. The value of θ is iteratively adjustedby a classical co-processor with the purpose of minimiz-ing the cost function

E[θ] =〈ψ(θ)|Hprob|ψ(θ)〉〈ψ(θ)|ψ(θ)〉

. (1)

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Numerous variational forms U(θ) have been explored inthe literature [18, 23–28]. Before introducing our ap-proach, in this section we briefly review two widely usedansatze highlighting their advantages and disadvantages.

A. Hardware-efficient Ansatz

The Hardware-efficient Ansatz (HEA), introduced inRef. [24], relies on gates that are native to the quantumhardware to produce circuits of high expressibility [29]and low depth. In particular, the HEA requires the ap-plication of successive blocks of parametrized single-qubitrotations followed by a generic entangling unitary UEnt..An example for N qubits is

UHEA(θ) =∏d

UEnt.

N∏n=1

R(n)Z (θZn,d)R

(n)Y (θYn,d), (2)

where θ = θZn,d, θYn,d collects all the variational pa-

rameters and R(n)σa (θ) = exp[−iθσa/2] denotes a single-

qubit rotation of angle θ around the a ∈ x, y, z axis onqubit n. σa is the corresponding Pauli matrix. The pa-rameter d is the number of layers, or depth, of the ansatz.Here and for the rest of this paper, we use the conven-

tion∏Ni Ui = UN · · · U1 for operator multiplication.

A feature of the HEA is that it is well suited to a broadexploration of the Hilbert space since it does not pur-posely favor a particular symmetry sector. This ansatzhas already been experimentally implemented to preparethe ground state of small molecules [24], to simulate thefolding of a few amino acid polymer [15], and to find thesolution of classical optimization problems [10]. How-ever, solving small instances of important problems doesnot provide a proof of scalability of the method for largersystems. Indeed, there is evidence that sufficiently ran-dom parametrized circuits, such as the ones producedby HEA, suffer from an exponentially vanishing gradientwith the number of qubits making them more difficult toconverge as the system size grows [30].

B. Variational Hamiltonian Ansatz

Ansatze that leverage the structure of the problem canavoid the aforementioned scalability issues since they donot explore the full exponentially large Hilbert space.Wecker et al. [23] introduced the Variational HamiltonianAnsatz (VHA), which consists of a parametrized adapta-tion of the quantum circuit implementing time evolutionunder the problem Hamiltonian via Trotterization. Inthe VHA framework, the state-preparation unitary reads

UVHA(θ) =∏d

∏j

eiθj,dHj , (3)

where θ = θj,d are the variational parameters

and Hprob =∑j Hj is the problem Hamiltonian ex-

pressed as the sum of non-commuting groups of terms

labeled Hj . The depth d is associated with each timeincrement of the Trotterization of the time-evolution op-erator. If grouping the terms is done efficiently, thisapproach can be implemented using few variational pa-rameters, therefore simplifying the classical optimization.However, depending on the complexity of the problem,circuits can be considerably longer as compared to thosetypically used with the HEA.

Fourier-transformed VHA (FT-VHA) To further re-duce the number of variational parameters, it is possi-ble to take advantage of the fact that most fermionicHamiltonians can be written as Hprob = T + V , where

the diagonal bases of T and V are related through thefermionic Fourier transformation (FT) [31–33]. With theFT-VHA variational form, the FT is used to alternate be-tween these bases at every Trotter step. In the contextof quantum chemistry, this is known as the split-operatormethod [34, 35]. This idea was also recently introducedby Babbush et al. [36] for the variational quantum simula-tion of materials. The state-preparation unitary therebyreads

UFT-VHA(τ ,ν)

=∏d

FT†

∏j

eiτj,dTj

FT

∏j

eiνj,dVj

,(4)

where τ = τj,d and ν = νj,d are the parameters asso-

ciated with T =∑j Tj = FT T FT† and V , respectively.

Since now both T and V are diagonal, they only containterms that commute and therefore the circuit decomposi-tion of their exponentials can be achieved exactly, whichwas not the case of T in the regular VHA. However, thiscomes at the cost of the long FT circuit [31, 33, 36].

Because they are built from the problem Hamiltonian,both VHA and FT-VHA respect the symmetries of theproblem. For example, if no term of Hprob allows thenumber of particles to change, this quantity will be con-served in the variational state |ψ(θ)〉. This choice re-stricts the variational search to a relatively smaller sub-space of the Hilbert space which, intuitively, can in-crease the performance of the VQA. Because of this, theVHA and FT-VHA ansatze are likely to perform betterthan HEA for large system sizes. However, as we showin Sect. IV, incorporating too much knowledge of theproblem can also be detrimental.

Another popular approach in quantum chemistry isthe UCCSD ansatz [3] which implements the exponentialof a set of single- and double-excitation operators. Al-though not strictly Hamiltonian-based, this method pre-serves the parity symmetry of fermions and conserves thenumber of particles. Despite potentially providing accu-rate results, the UCCSD ansatz circuits can be very deep,limiting its applicability on near-term quantum devices.

3

II. QUANTUM-OPTIMAL-CONTROL-INSPIRED ANSATZ

(QOCA)

To address the drawbacks of the ansatze discussedabove, we propose an ansatz that borrows ideas from thetheory of quantum optimal control [37–40], and whichwe therefore dub the Quantum-Optimal-Control-inspiredAnsatz, or QOCA. The main idea behind QOCA re-sides in the introduction of carefully chosen symmetry-breaking unitaries into the symmetry-preserving ansatzVHA. In this section, we begin by reviewing some of thecentral aspects of the theory of quantum optimal control,and then show how these ideas can be incorporated in thedesign of variational forms.

A. Quantum optimal control

Quantum optimal control (QOC) theory describes themethods to optimally steer a quantum system from aninitial state to a known final state [41]. Such techniqueshave been applied to a wide variety of problems includingthe quantum control of chemical reactions [42, 43], spinsin nuclear magnetic resonance experiments [39, 44] and,more recently, to superconducting qubits [40, 45].

In this approach, the control Hamiltonian is spec-ified by a set of time-independent drive Hamiltoni-ans Hk whose amplitudes are parametrized by thetime-dependent coefficients ck(t) ∈ R. The total

Hamiltonian H(t) is then, in general, time-dependentsuch that

H(t) = H0 +∑k

ck(t)Hk, (5)

with H0 the free, or drift, Hamiltonian of the controlledsystem. Solving the Schrdinger equation of the drivensystem results in the unitary U(t), which can propagate

pure states through time as |ψ(t)〉 = U(t)|ψ(0)〉.The system described by the Hamiltonian of Eq. (5),

defined in a Hilbert space of dimension n, is said to becontrollable if U(t) can be any matrix of SU(n). Inother words, the system is controllable if for any ini-tial state |ψ(0)〉, there exists a set controls ck(t) and atime T > 0 for which the state |ψ(T )〉 can be any targetstate of the Hilbert space [41].

Quantum optimal control techniques, such as theGRAPE algorithm [39], provide a method for designingthe control pulses ck(t) to achieve a desired state prepa-ration. This is usually realized by seeking the set of con-trols and time T that optimize a cost function charac-terizing the state-preparation fidelity, which may includeconstraints such as the control time and the maximumpulse amplitudes.

In the GRAPE algorithm, time is discretized into Nincrements, or pixels, of duration ∆t such that the to-tal evolution occurs in a time T = N∆t. Using this

discretization, the continuous control fields ck(t) arenow parametrized by the new constant piecewise controlfields uk = uk,j as

ck(t) =

N−1∑j=0

uk,j uj (t,∆t), (6)

where uj(t,∆t) ≡ Θ(t− j∆t)−Θ(t− (j + 1)∆t) with Θthe Heaviside function. The time evolution operator fora time T therefore reads

U(T ) =

N−1∏j=0

exp

[−i∆t

(H0 +

∑k

uk,jHk

)], (7)

and optimality is achieved by iteratively tuning the val-ues of the discrete control fields uk,j. Because this

time propagator incorporates drive terms Hk, that typ-

ically do not commute with the drift Hamiltonian, U(T )may implement unitary operations that are distinct fromthat generated by the drift Hamiltonian alone. In stan-dard QOC problems, fast and efficient optimization ofthe control fields is possible because the target state (oroperation) is known. This is, however, not the case inthe context of VQA. Adapting these techniques to theVQA setting therefore requires to eliminate any informa-tion about the target state from the QOC cost function,therefore making the optimization less straightforward.

B. The QOCA variational form

Building on the concept of quantum optimal control,we modify the VHA by constructing a variational formwhich includes a set of drive terms Hk in addition to

the problem Hamiltonian Hprob. QOCA therefore mimicstime evolution under the new Hamiltonian

HQOCA(t) = Hprob +∑k

ck(t)Hk, (8)

where, by design, [Hprob, Hk] 6= 0 ∀ k. We then constructthe state-preparation circuit for QOCA by parameteriz-ing the time-evolution-like operator

UQOCA(θ, δ) =∏d

∏j

eiθj,dHj∏k

eiδk,dHk

, (9)

where Hprob =∑j Hj and θ = θj,d are the prob-

lem Hamiltonian parameters. Similar to Eq. (6), δ =δk,d are the discrete drive amplitudes of the controlfields ck(t) of Eq. (8) which we use as variational param-eters. Again, d is the depth of the ansatz and is analogto the steps in the time evolution.

A key concept of QOCA is that the problem Hamil-tonian part helps constraining the variational search tothe relevant symmetry sector of the Hilbert space, while

4

the drive part allows the ansatz to take shortcuts by tem-porarily exiting this sector. This concept is schematicallydrawn on Fig. 1b where we illustrate possible paths in theHilbert space for the HEA, VHA and QOCA variationalforms.

In principle, one has the freedom to select any driveHamiltonians that do not commute with Hprob. However,it is not straightforward to predict which choice will havethe most positive impact on the outcome of the VQA.One option is to use an adaptive approach such as the onedescribed in Refs. [46, 47]. However, in the next sectionwe show how simple considerations can help to bound thenumber of interesting drive operators, and suggest whichof these could be more effective.

C. Which drive Hamiltonians are useful forfermions?

With the objective of applying QOCA to theFermi-Hubbard model, we consider the time-dependentfermionic Hamiltonian

Hf (t) =∑j

(αj(t)aj + α∗j (t)a†j)

+∑i,j

βij(t)(a†i aj + a†j ai) +

∑i,j

γij(t)a†i aia

†j aj ,

(10)

where a†j and aj are fermionic ladder operators ofspin-orbital j respecting the anti-commutation rela-

tions ai, a†j = δij and ai, aj = a†i , a†j = 0. Impor-

tantly, Hf (t) is controllable in the sense that any unitarymatrix can be generated by solving its Schrdinger equa-tion [48, 49].

We note that while the first term of Hf (t) is unphysi-cal since it breaks the parity symmetry of fermions, thequadratic and quartic terms occur in many physical mod-els. This makes Hf (t) attractive for designing drivenphysically inspired ansatze as we are guaranteed thatdrive terms of form α(t)a + α∗(t)a† will not commutewith the physical problem Hamiltonian. Interestingly,the use of such terms has been proposed in the contextof variational error suppression [4] as they may allow avariational state to re-enter a particular symmetry sectorto correct for the effect of symmetry-breaking errors.

In the QOCA variational form, we propose to firstwrite Hf (t) keeping only the quadratic and quartic terms

that also appear in Hprob, along with few additionalsymmetry-breaking drive terms. As in Eq. (9), we thenparametrize the resulting time-evolution-like operator us-ing the associated α(t), β(t), and γ(t) coefficient as pa-rameters. With these choices, the QOCA variationalform generates circuits that are only slightly differentfrom those generated by the problem Hamiltonian.

We also note that the principles of this analysis canbe extended to the simulation of non-fermionic Hamil-tonians, provided a controllable Hamiltonian for thesesystems.

III. QOCA FOR THE FERMI-HUBBARDMODEL

For completeness, we start this section by review-ing the Fermi-Hubbard model and explain how we usethe QOCA variational form to prepare its ground state.We motivate our choice of initial state, and elaborateon the selection and circuit decomposition of the driveterms. Finally, we introduce short-QOCA, a variantof QOCA that yields shorter circuits by dropping someterms of Hprob from the Hamiltonian that generates theregular QOCA ansatz.

A. The Fermi-Hubbard model (FHM)

The Fermi-Hubbard model is an iconic model in thestudy of strongly correlated materials [50]. It describesinteracting spin- 12 fermions on a lattice where each sitecan be occupied by up to two particles of opposite spins.The Hamiltonian of the FHM for L lattice sites takes theform

HFHM = −t∑〈i,j〉,σ

a†iσajσ

≡ T

+U

L∑i=1

ni↑ni↓ − µ∑i,σ

niσ

≡ V

,

(11)where i, j are the lattice-site indices, and σ = ↑, ↓ la-bels the spin degree of freedom. In the first term, 〈i, j〉 de-

notes a sum over nearest-neighbor sites, and niσ = a†iσaiσis the occupation operator of the spin-orbital labeled iσ.

The first term of Eq. (11) represents hopping betweenneighboring sites with amplitude −t and will generallybe referred to as T . This term is diagonal in momentumspace if periodic boundary conditions are used, and itsground state consists of delocalized plane waves. Thesecond term is a non-linear, on-site Coulomb repulsion ofstrength U , while the last term is the chemical potential.These last two terms are diagonal in the position basisand, taken together, are denoted V . The ground state ofV is described by wave functions localized on the sites.

A particularly interesting instance of the FHM is thehalf-filling regime (which occurs for µ = U/2) at inter-

mediate coupling, U/t ∼ 4. In this regime, both T and Vcontribute significantly to the system’s energy, thus cre-ating competition between the localized and delocalizedstates of the electrons, leading to rich physics such as theMott transition. Because it becomes impossible to accu-rately treat either part of the Hamiltonian perturbatively,numerical exact diagonalization of the FHM is difficultbeyond 24 lattice sites at half-filling [51]. As we seek tobenchmark the usefulness of our variational form for allcases, we work in this particularly challenging regime.

Despite its apparent simplicity, this model has beenused to study systems ranging from heavy fermions [52]to high-temperature superconductors [53, 54]. As a re-sult, it is an interesting problem to benchmark near-term

5

ei ν Vei τFT FT†

FT FT†

↑↓ ei τ

FT-VHA:

ei ν VD

D

↑↓

short-QOCA (this work):

QOCA (this work):

ei ν Vei τ T D

D

↑↓ ei τ T

VHA:

ei ν V↑↓ ei τ T

ei τ T

HEA:RY RZ

RY RZ

↑↓

Symmetry of the Ansätze’s structure

Arbitrary Problem-inspired

|ψ0⟩

|Ω⟩

particlesN

Hilbert space

HEA

QOCA

VHA

a b

FIG. 1. a Single circuit layer of the ansatze studied in this work arranged by the symmetry of their structure. A highsymmetry means that the ansatz is completely built around the problem Hamiltonian while a low one reflects the arbitrarinessof its circuit. We show the hardware-efficient Ansatz (HEA) Eq. (2), the variational Hamiltonian Ansatz (VHA) Eq. (3), theFourier-transformed VHA (FT-VHA) Eq. (4), the Quantum-Optimal-Control-inspired Ansatz (QOCA) Eq. (9) along with ashallower version of QOCA, the short-QOCA ansatz Eq. (18). The horizontal lines represent the qubit registers that encodethe spin orbitals associated with the ↑ or ↓ spins. For HEA, the entangling block is a ladder of CNOT similar to the ones inthe Notation box of Figure 2. For all other ansatze, T and V are respectively the kinetic and interaction parts of the problemHamiltonian and τ ,ν are their associated variational parameters. For FT-VHA, we have that T = FT T FT†. The drivecircuit D is defined in Eq. (17) and illustrated in Figure 2. b Possible paths in the Hilbert space for the HEA, VHA and QOCAvariational forms. The initial state |ψ0〉 and the target state |Ω〉 are in the same symmetry sector containing N particles. Since

HEA does not conserve the symmetries of H, its path easily escapes from the fixed particle number subspace, while VHAis restricted to it. By introducing symmetry-breaking terms, QOCA has the ability to escape slightly from the N particlessubspace to find shortcuts in Hilbert space.

quantum computers [55], and a useful performance testfor variational ansatze. For these reasons, variationalquantum algorithms have already been used to find theground state of the FHM, for example using the HEAvariational form [56], the VHA [21–23, 57, 58], and othersymmetry-preserving ansatze [21, 25, 55, 59, 60].

B. Encoding and parametrization of the ansatze

We use the Jordan-Wigner (JW) transformation to en-code fermionic Fock states into qubits registers, as de-tailed in Appendix A. Moreover, we work in real spaceand order the basis vectors for the 2L spin orbitalsas |f1↑ . . . fL↑; f1↓ . . . fL↓〉 with fp ∈ 0, 1 the occupa-tion of orbital p.

Using this purely conventional choice, in Fig. 1a weschematically draw one layer of the circuits implementingthe different ansatze discussed above and arranged bythe symmetry of their structure. A highly symmetricansatz is completely built around Hprob while a weaklysymmetric construction is arbitrary with respect to theproblem.

To parametrize these circuits, we consider two possiblestrategies: one corresponding to full parametrization ofthe single- and two-qubit gates and the other having anumber of parameters that only grows with the depth ofthe ansatz, but not with the number of qubits. Wheneverused, the latter is specified with the label ‘scalable’. Bothstrategies are elaborated on in Appendix B and details ofthe numerical simulation are presented in Appendix C.

C. Initial state

In general, the performance of VQAs strongly dependson the choice of initial state and variational parameters.The initial state acts as an educated guess to the targetstate and is often chosen such as to be easily computableclassically. Moreover, because the initialization stage ofa variational algorithm should be straightforward or oth-erwise be treated as a separate routine [61], we are in-terested in benchmarking the performance of the QOCAvariational form for the simple initial state

|ψ0〉 = H⊗N |0〉 = |+〉⊗N , (12)

where H is the Hadamard gate. In addition to being easyto prepare, this initial state corresponds to half-fillingand zero total spin, placing it in the same symmetry sec-tor as the target state.

While this choice allows us to demonstrate the useful-ness of the QOCA variational form given unstructured,simple initial conditions, we also show how the conver-gence can be improved further by using the ground stateof the non-interacting FHM fixing U = µ = 0 in Eq. (11)as initial state. More details on how to prepare this morecomplex state are provided in Appendix D.

6

D. Drive Hamiltonians

With the goal of reducing the number of variationalparameters, we fix αj(t) to 1 and i in Eq. (10) leading to

H1 =

L∑j=1

(a†j + aj), (13)

H2 =

L∑j=1

i(a†j − aj). (14)

We moreover obtain the drive equations for a spinlesssystem and independently apply the resulting circuit tothe two subspaces corresponding to the spin projectionsup and down for all sites. Performing the JW transfor-mation on Eqs. (13) and (14), we find

H1 7→L∑j=1

Xj

⊗l<j

Zl, (15)

H2 7→L∑j=1

Yj⊗l<j

Zl, (16)

where X, Y and Z are Pauli matrices. To incor-porate these expressions into the QOCA variationalform Eq. (9), we perform a first-order Trotter-Suzuki de-composition, arriving at the circuit equation for the dthlayer of the ansatz,∏k=1,2

eiδk,dHk

≈L∏j=1

exp

iδ1,d Xj

⊗l<j

Zl

exp

iδ2,d Yj ⊗l<j

Zl

,(17)

where δk,d are the variational parameters associatedwith the kth drive term of that layer. A schematic of thecircuit implementing Eq. (17) for 4 qubits is illustratedin Fig. 2 where we also show a compiled version in termsof CNOTs.

E. The short-QOCA variational form

One drawback of QOCA is that, depending on the formof the drive, the corresponding quantum circuits can belong. Here we demonstrate a practical approach for thereduction of the circuit depth without compromising theperformance.

Because the drive D in Fig. 2 and the kinetic part ofthe FHM Eq. (11) are both block-diagonal in the spindegree of freedom, we chose to remove the latter term,which is also costly in terms of two-qubit gates, arrivingto the simplified form of the ansatz

UsQOCA(ν, δ) =∏d

∏j

eiνj,dVj∏k

eiδk,dHk

, (18)

RY RX ZY

ZX Z

Y

ZZX

Z

ZY

ZZ

ZX

ZZ

D ≡a

b

c≡Notation:

=RY RX Z

YZX Z

YZX Z

YZX

HEA:

VHA:

ei ν Vei τ TkFT FT†

ei τ TkFT FT†

↑↓

FT-VHA:

QOCA:

ei ν VD

D

↑↓

short-QOCA:

ei ν Vei τ T D

D

↑↓ ei τ T

ei ν V↑↓ ei τ T

ei τ T

RY RZ

RY RZ

↑↓

⋮X

Z≡

H RZ H

exp [i θ Z… ZX] ⋮Y

Z≡

G RZ G

exp [i θ Z… Z Y]

FIG. 2. a Circuit decomposition of the drive Eq. (17) usedfor QOCA. This circuit generalizes to any number of qubits byappending more Z . . . ZY and Z . . . ZX multi-qubit gates atthe end. We also show the circuit compiled to one- and two-qubit (CNOT) gates. b shows a decomposition of the multi-qubit gates based on a conventional approach to decomposeexponentials of Pauli strings into circuits of CNOTs describedin [62]. The transformationH = (X+Z)/

√2 is the Hadamard

gate which changes between the X and Z bases and G = (Y +

Z)/√

2 is the equivalent transformation between the Y and Zbases. The angles of the rotations Ra(θ) = exp[−iθσa/2]are the variational parameters, where σa is a Pauli matrix.c shows the notation shortcut used for the ladders of CNOTs.

where V =∑j Vj is the on-site interaction part of the

Fermi-Hubbard Hamiltonian Eq. (11) and ν = νj,d arethe associated variational parameters. We refer to thissimplified version of the QOCA variational form as short-QOCA, see Fig. 1.

IV. NUMERICAL RESULTS

In this section, we compare results obtained from nu-merical simulations of QOCA and short-QOCA for theFermi-Hubbard model, and contrast these results withthose obtained with the other ansatze discussed in thisarticle. As an illustration of the use of QOCA beyond theFermi-Hubbard model, we also present a comparison ofthe performance of this ansatz over a hardware-efficientapproach and the UCCSD ansatz for a 12-qubit repre-sentation of the H2O molecule.

Throughout this section, we use the fidelity with re-spect to the target state |Ω〉 (i.e. ground state of theFHM or of the water molecule) as defined by

Fidelity = |〈ψ(θ)|Ω〉|2, (19)

7

10 4

10 2

100Fermi-Hubbard model

0 2 4 6 8 10Number of Ansatz layers d

10 3

10 2

10 1

100

HEAVHAFT-VHA

QOCAQOCA (scalable)short-QOCA

1Fi

delit

y

2 × 2

2 × 3

FIG. 3. Final variational state infidelities with respect tothe target state as a function of the number of layers d of thevariational forms of this work. Top panel is for a 2×2 plaquet-te while the bottom panel is a 2× 3 system without periodicboundary conditions. The initial state is |+〉⊗N for all cases.Data at d = 0 corresponds to the initial state alone, whichhas a fidelity of 0.035 with the target state. Unless specified,all ansatze are fully parametrized according to Appendix B 1.

to quantify the quality of the variational state |ψ(θ)〉.

A. Fermi-Hubbard model

We consider 2 × 2 (8 qubits) and 2 × 3 (12 qubits)lattices of the Fermi-Hubbard model at half-filling withopen boundary conditions. We note that the former con-figuration can also be seen as a periodic 1 × 4 chain.This allows us to compare with the FT-VHA variationalform, as the fermionic Fourier transform on which thisapproach relies is defined for periodic boundary condi-tions. Importantly, we find that for smaller systems suchas the four-qubits 2× 1 dimer, all ansatze converge in afew tens of iterations on the ground-state energy with aprecision of < 10−7 using a single ansatz layer, d = 1,except for the HEA which requires two layers.

a. Comparing the ansatze For systems with fourand six fermionic sites, we observe important variationsin the ability of the different ansatze to converge to theground state energy. This is illustrated in Fig. 3 whichshows, for all ansatze, the final state infidelity as a func-tion of the number of ansatz layers, d, initialized withthe simple half-filled state of Eq. (12). The maximumfidelities achieved for all ansatze are reported in Table Ialong with resource counts using a circuit compilation interms of CNOTs.

We first note that VHA and FT-VHA perform poorlyfor both system sizes and that their performance does

TABLE I. Maximum fidelities with respect to the ground stateof the FHM, attained for d ansatz layers, each requiring anumber nθ/d of variational parameters and nCX/d CNOTsper layer. The latter estimate assumes an all-to-all connectiv-ity and the same compiling procedure is used for all ansatze.

Hubbard model Max Fid. d nθ/d nCX/d

2×

2

(8qubit

s)

HEA 0.9876 9 16 7VHA 0.1343 8 8 56FT-VHA 0.1315 7 8 120QOCA 0.9999 4 16 88QOCA (scalable) 0.9992 10 5 88short-QOCA 0.9999 9 12 40

2×

3

(12

qubit

s) HEA 0.7276 10 24 11VHA 0.0804 10 13 116QOCA 0.9965 9 25 172QOCA (scalable) 0.8822 10 6 172short-QOCA 0.7476 8 18 68

not improve with the addition of more entangling layers,i.e. increasing d. Because these ansatze are particle-number conserving, this observation suggests that VHAand FT-VHA may not efficiently search over all states offixed particle number in the variational landscape, as wasoriginally proposed. Moreover, since FT-VHA performssimilarly to VHA for the 2× 2 system, we also concludethat alternating bases with the fermionic Fourier trans-form does not yield superior results for these lattice sizes.

Interestingly, QOCA systematically reaches theground state of the Fermi-Hubbard model with signifi-cantly more accuracy than VHA for both system sizes,indicating that the additional symmetry-breaking termshelp the convergence. This advantage persists even whendrastically reducing the number of variational parame-ters from 16 to 5 per layer in the case of the scalableparametrization of QOCA, which converged with 0.9992fidelity at d = 10 for the 2 × 2 system. The hardware-efficient approach also performs better than VHA, al-though it uses considerably more parameters than allother ansatze given it generally requires more layers toachieve similar performances. It is unclear how one mightreduce that number to a favorable scaling.

Data obtained with the short-QOCA variational formshow that the QOCA circuits can be substantially short-ened by removing more than half of the two-qubit gatesat every step without much compromise on the perfor-mance for small systems. In fact, for the 2× 2 Hubbardmodel, a fidelity of 0.9999 is achieved with 9 layers of thisansatz.

With improved fidelities for shallower circuits whichuse fewer variational parameters than standard ap-proaches, we find that QOCA provides significant gainwith respect to other common ansatze.

b. The benefits of breaking symmetries Figure 4shows the evolution of the average number of particlesper lattice site (top panel) and the infidelity of the vari-ational state with respect to the target state (bottom

8

0.95

1.00

1.05

101 102 103 104 105

Optimizer iterations10 4

10 2

100

1Fi

delit

y

HEAVHAQOCAQOCA (scalable)

NL

FIG. 4. Top: Average number of particles per lattice sitein the variational state at every iteration of the VQA rou-tine. 〈N〉 =

∑i,σ〈niσ〉 is the total occupation and L is the

number of sites. Bottom: corresponding variational state in-fidelity, 1 − |〈ψ(θ)|Ω〉|2, with respect to the ground state ofthe Fermi-Hubbard model |Ω〉. The results are for a 2 × 2system and the initial state is |+〉⊗N for all ansatze. Runsfor ansatz depth d = 9 was used for HEA and d = 10 for theothers, but this behavior is observed for most d.

panel) throughout the optimization process for the samesimulations as in Fig. 3.

Focusing first on the top panel we first note that, be-cause the initial state |+〉⊗N is half-filled, all variationalstates begin in the correct particle-number symmetry sec-tor of the Hilbert space with 〈N〉/L = 1. Because VHAdoes not contain terms that allow the particle number tochange, this quantity is observed to be constant through-out the optimization. We hypothesize that the poor per-formance of this ansatz in reaching the ground state iscaused by the inability of this variational form to over-come local minima in parameter space.

In contrast, with their particle-non-conserving driveterms, both parametrizations of QOCA allow the averagesite occupancy to deviate from 〈N〉/L = 1 as the driveangles are being tuned away from zero by the optimizer.As seen in Fig. 4, this can lead to the sharp features ob-served in the first few ∼ 102 iterations as the classicaloptimizer can initially overweight the value of individualterms. Over the full optimization, the number of particlesdeviates only slightly from the target value 〈N〉/L = 1with changes of only ∼ 5% of the site occupancy. Thisis an indication that the symmetry-breaking terms inQOCA allow the ansatz to explore a Hilbert space that isslightly larger than the manifold of fixed particle number.Nevertheless, we find that these relatively small excur-sions out of the target symmetry sector can significantlyease convergence of the VQA. Indeed, we observe thatthe onset of the return to the target symmetry sector, asindicated by the vertical dashed lines in Fig. 4 is often as-sociated with the abrupt descents in the infidelity, whichmay indicate that regions of steep gradients in parameterspace are found.

This behavior is also observed for the hardware-

0 5 100.0

0.2

0.4

0.6

0.8

1.0

Fide

lity

VHA

0 5 10

QOCA

| + N

Selected Tground stateSuperposition ofT ground states

Number of Ansatz layers d

Initial states

FIG. 5. Variational state fidelities with respect to the groundstate of the 2× 2 FHM as a function of the number of ansatzlayers, d, for the VHA and QOCA variational forms. Re-sults with three initial states are presented: (solid) Hadamardgates on every qubit |+〉⊗N , (dotted) a selected ground state

of T corresponding to |Ω(1)T 〉 of Appendix D, and (dashed)

a superposition of ground states of T corresponding to |ΩT 〉of Appendix D.

efficient ansatz of Eq. (2) which also, does not preserve

the symmetries of Hprob. This phenomenon is not par-ticular to the realizations displayed in the figure, and itis also observed for other system sizes and initial states.

We note, however, that these desired regions in pa-rameter space would never be found if an error-miti-gation technique based on symmetry verification wereemployed [63, 64]. Indeed, in these schemes the varia-tional states are post-selected after the energy measure-ments only if they conserve desired symmetries of thetarget state. However, other strategies for error mitiga-tion remain applicable [65–67].

c. Initial state Because it provides a simple settingto benchmark the performance of the different ansatze,we have so far considered only the single easily-preparedinitial state of Eq. (12). Improved approximation to theground state can, however, be obtained if a more struc-tured initial state is considered although at the price ofmore complex state preparation circuits.

In Fig. 5, we compare the performance of the VHAand QOCA variational forms on the 2 × 2 lattice withthe following initial states of increasing complexity: i)the simple state |+〉⊗N , ii) one of the degenerate ground

states of T labelled |Ω(1)T 〉 in Appendix D, and iii) the

superposition of ground states of T labelled |ΩT 〉 of Ap-pendix D.

While the final variational state obtained with VHAstrongly depends on the initial state, QOCA systemati-cally achieves convergence with fidelity > 0.9999, regard-less of the initialization choice. Again because of its abil-ity to move between symmetry sectors, these results illus-trate QOCA’s robustness to simple, unstructured, initialconditions that can have very small overlaps with the tar-

9

TABLE II. Maximum fidelities obtained with d entanglinglayers, nθ/d variational parameters and nCX/d CNOTs perlayer and different initial states for the QOCA, HEA andUCCSD variational forms applied to the water molecule. Theinitial states are either the Hartree-Fock (HF) approximationto the ground state or the equal superposition of all basisstates |+〉⊗N . Again, the gate count estimate assumes an all-to-all connectivity and the same compiling procedure is usedfor all ansatze.

Water molecule (12 qubits)

Initial state Max Fid. d nθ/d nCX/d

QOCA |+〉⊗N 0.9742 1 23 108|+〉⊗N 0.9931 5 23 108

HF 0.9735 1 23 108HF 0.9917 7 23 108

HEA |+〉⊗N 0.9820 8 24 11UCCSD HF 0.9748 1 8 528

get ground state. For the two variational forms, using asuperposition of the degenerate ground states of T as ini-tial state (dashed lines) leads to convergence with fewerentangling layers. This, however, comes at the cost of sig-nificantly increasing the complexity of the initializationstage of the VQA (see Appendix D).

B. Proof-of-principle implementation of the H2Omolecule

The previous section illustrates how QOCA can ap-proximate the ground state of the FHM with systemati-cally more accuracy than other ansatze even when facedwith unstructured initial conditions. In order to inves-tigate the broader applicability of this method, we nowbenchmark the QOCA variational form on a quantumchemistry problem. As a proof-of-principle test, we con-sider the H2O molecule in its equilibrium configuration.Because we freeze the core orbitals, this problem maps to12 qubits using the STO3G basis set. The Hamiltonianis obtained using the PySCF driver as provided in QiskitChemistry [68]. We compare the performance of QOCAagainst HEA together with the well-known chemistry-inspired UCCSD ansatz [3]. Because the Hamiltonian de-scribing the water molecule has significantly more termsthan the FHM, directly implementing Hamiltonian-basedansatze as it is done above would lead to very long cir-cuits. Therefore, we do not consider VHA for this prob-lem.

In consequence, as a simple implementation of QOCAto a quantum chemistry problem, we use a variationof the ansatz based on the 12-qubit Hamiltonian of anopen 1 × 6 Fermi-Hubbard chain with the drive termsof Eqs. (13) and (14). Although the water moleculeHamiltonian describes a richer set of fermionic interac-tions than the FHM, this choice of ansatz offers one ofthe simplest construction that simulates electron-electroncorrelations and is therefore a good starting point. More-

over, the ansatz is fully parametrized as before and thesimulations were achieved under the same numerical con-ditions.

The maximum fidelities achieved for the QOCA, HEAand UCCSD variational forms are reported in Table IIfor different number of ansatz layers d and initial states,which are either the Hartree-Fock (HF) approximationto the ground state or the equal superposition of all ba-sis states |+〉⊗N . Both initial states require one layer ofsingle-qubit gates to prepare. The number of variationalparameters (nθ/d) and CNOT gates (nCX/d) per layerare also presented. For the case of UCCSD we use d = 1,as it is proven to be enough for the simulation of chemi-cal systems [26, 69]. For this reason, UCCSD uses signifi-cantly fewer parameters than other approaches, however,the resulting circuit requires roughly the same two-qubit-gate count as a d = 5 QOCA circuit.

We observe that a single layer (d = 1) QOCA cir-cuit can prepare the ground state of the water moleculewith fidelity 0.9742, a performance which is comparableto that of the well-established UCCSD ansatz (0.9748).However, while this does not improve the performanceof UCCSD, adding layers up to d = 5 for QOCA in-creased the fidelity to 0.9931. Interestingly, the |+〉⊗Ninitial state, which has a 9.95 × 10−5 overlap with thetarget state, yields better results for QOCA with feweransatz layers than the Hartree-Fock initial state, whichhas a 0.9735 overlap. These simulations suggest thatQOCA can be useful also for quantum-chemistry prob-lems. Modifying the QOCA circuit to better reproducethe interactions between the spin orbitals of the watermolecule could lead to further improvements in perfor-mance.

V. CONCLUSION

We introduced the Quantum-Optimal-Control-ins-pired Ansatz by adding carefully chosen symmetry-breaking drive terms to the problem Hamiltonian andparametrizing the resulting time-evolution-like opera-tor. We first applied QOCA to the half-filled Fermi-Hubbard model and found that in most cases it yieldsto a faster and more accurate convergence than standardapproaches, even with unstructured initial states havinglittle overlap with the target ground state. We showedevidence that this improved convergence is made possi-ble by the symmetry-breaking terms allowing for smallexcursions outside of the target symmetry sector of theproblem Hamiltonian. Moreover, we used QOCA to pre-pare the ground state of the water molecule, and showedthat it can surpass the commonly used UCCSD ansatzwith drastically shorter circuits.

Its broader applicability and the flexibility in choos-ing drive terms make QOCA a promising approach totackle a wide range of quantum chemistry and materialsproblems on near-term quantum computers. Althoughthe QOCA circuits are currently too deep to be imple-

10

mented reliably on today’s quantum devices, we expectthat it may exhibit some resilience to symmetry-breakingerrors. Our work represents a first step towards the de-velopment of a more general class of symmetry-breakingansatze for variational quantum algorithms.

ACKNOWLEDGMENTS

We thank David Poulin, Jonathan Gross and Alexan-dre Daoust for useful discussions. This work was un-

dertaken thanks in part to funding from NSERC, theCanada First Research Excellence Fund and the U.S.Army Research Office Grant No. W911NF-18-1-0411.

Note added After completion of this work, we becameaware of related work that was recently posted [70].

[1] J. Preskill, Quantum computing in the nisq era and be-yond, Quantum 2, 79 (2018).

[2] K. Wright, K. Beck, S. Debnath, J. Amini, Y. Nam,N. Grzesiak, J.-S. Chen, N. Pisenti, M. Chmielewski,C. Collins, et al., Benchmarking an 11-qubit quantumcomputer, Nature Communications 10, 1 (2019).

[3] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q.Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. Obrien,A variational eigenvalue solver on a photonic quantumprocessor, Nature communications 5, 4213 (2014).

[4] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New Journal of Physics 18, 023023(2016).

[5] C. Zoufal, A. Lucchi, and S. Woerner, Quantum genera-tive adversarial networks for learning and loading randomdistributions, arXiv preprint arXiv:1904.00043 (2019).

[6] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi,L. Cincio, and P. J. Coles, Variational quantum linearsolver: A hybrid algorithm for linear systems, arXivpreprint arXiv:1909.05820 (2019).

[7] M. Schuld and N. Killoran, Quantum machine learningin feature hilbert spaces, Physical review letters 122,040504 (2019).

[8] V. Havlıcek, A. D. Corcoles, K. Temme, A. W. Harrow,A. Kandala, J. M. Chow, and J. M. Gambetta, Super-vised learning with quantum-enhanced feature spaces,Nature 567, 209 (2019).

[9] E. Farhi, J. Goldstone, and S. Gutmann, A quan-tum approximate optimization algorithm, arXiv preprintarXiv:1411.4028 (2014).

[10] P. K. Barkoutsos, G. Nannicini, A. Robert, I. Tavernelli,and S. Woerner, Improving variational quantum opti-mization using cvar, Quantum 4, 256 (2020).

[11] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, andM. Troyer, Elucidating reaction mechanisms on quantumcomputers, Proceedings of the National Academy of Sci-ences 114, 7555 (2017).

[12] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark,C. Nayak, and M. Troyer, Solving strongly correlatedelectron models on a quantum computer, Physical Re-view A 92, 062318 (2015).

[13] J. Olson, Y. Cao, J. Romero, P. Johnson, P.-L.Dallaire-Demers, N. Sawaya, P. Narang, I. Kivlichan,M. Wasielewski, and A. Aspuru-Guzik, Quantum infor-mation and computation for chemistry, arXiv preprintarXiv:1706.05413 (2017).

[14] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. John-son, M. Kieferova, I. D. Kivlichan, T. Menke, B. Per-opadre, N. P. Sawaya, et al., Quantum chemistry in theage of quantum computing, Chemical reviews 119, 10856(2019).

[15] A. Robert, P. K. Barkoutsos, S. Woerner, and I. Tav-ernelli, Resource-efficient quantum algorithm for proteinfolding, arXiv preprint arXiv:1908.02163 (2019).

[16] A. Di Paolo, P. K. Barkoutsos, I. Tavernelli, andA. Blais, Variational quantum simulation of ultrastronglight-matter coupling, arXiv preprint arXiv:1909.08640(2019).

[17] B. T. Gard, L. Zhu, G. S. Barron, N. J. Mayhall, S. E.Economou, and E. Barnes, Efficient symmetry-preservingstate preparation circuits for the variational quantumeigensolver algorithm, npj Quantum Information 6, 1(2020).

[18] P. K. Barkoutsos, J. F. Gonthier, I. Sokolov, N. Moll,G. Salis, A. Fuhrer, M. Ganzhorn, D. J. Egger, M. Troyer,A. Mezzacapo, et al., Quantum algorithms for electronicstructure calculations: Particle-hole hamiltonian and op-timized wave-function expansions, Physical Review A 98,022322 (2018).

[19] R. Sagastizabal, X. Bonet-Monroig, M. Singh, M. Rol,C. Bultink, X. Fu, C. Price, V. Ostroukh, N. Muthusub-ramanian, A. Bruno, et al., Error mitigation by sym-metry verification on a variational quantum eigensolver,arXiv preprint arXiv:1902.11258 (2019).

[20] M. Ganzhorn, D. J. Egger, P. Barkoutsos, P. Ollitrault,G. Salis, N. Moll, M. Roth, A. Fuhrer, P. Mueller, S. Wo-erner, et al., Gate-efficient simulation of molecular eigen-states on a quantum computer, Physical Review Applied11, 044092 (2019).

[21] C. Cade, L. Mineh, A. Montanaro, and S. Stanisic,Strategies for solving the fermi-hubbard modelon near-term quantum computers, arXiv preprintarXiv:1912.06007 (2019).

[22] A. Montanaro and S. Stanisic, Compressed variationalquantum eigensolver for the fermi-hubbard model, arXivpreprint arXiv:2006.01179 (2020).

[23] D. Wecker, M. B. Hastings, and M. Troyer, Progress to-wards practical quantum variational algorithms, PhysicalReview A 92, 042303 (2015).

[24] A. Kandala, A. Mezzacapo, K. Temme, M. Takita,M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for smallmolecules and quantum magnets, Nature 549, 242

11

(2017).[25] P.-L. Dallaire-Demers, J. Romero, L. Veis, S. Sim, and

A. Aspuru-Guzik, Low-depth circuit ansatz for prepar-ing correlated fermionic states on a quantum computer,Quantum Science and Technology 4, 045005 (2019).

[26] J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J.Love, and A. Aspuru-Guzik, Strategies for quantum com-puting molecular energies using the unitary coupled clus-ter ansatz, Quantum Science and Technology 4, 014008(2018).

[27] I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney,A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush,Quantum simulation of electronic structure with lin-ear depth and connectivity, Physical review letters 120,110501 (2018).

[28] A. J. Woitzik, P. K. Barkoutsos, F. Wudarski, A. Buch-leitner, and I. Tavernelli, Entanglement production andconvergence properties of the variational quantum eigen-solver, arXiv preprint arXiv:2003.12490 (2020).

[29] S. Sim, P. D. Johnson, and A. Aspuru-Guzik, Express-ibility and entangling capability of parameterized quan-tum circuits for hybrid quantum-classical algorithms, Ad-vanced Quantum Technologies 2, 1900070 (2019).

[30] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Bab-bush, and H. Neven, Barren plateaus in quantum neuralnetwork training landscapes, Nature communications 9,4812 (2018).

[31] F. Verstraete, J. I. Cirac, and J. I. Latorre, Quantum cir-cuits for strongly correlated quantum systems, PhysicalReview A 79, 032316 (2009).

[32] A. J. Ferris, Fourier transform for fermionic systems andthe spectral tensor network, Physical review letters 113,010401 (2014).

[33] Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy,and S. Boixo, Quantum algorithms to simulate many-body physics of correlated fermions, Physical Review Ap-plied 9, 044036 (2018).

[34] J. A. Fleck, J. Morris, and M. Feit, Time-dependent prop-agation of high energy laser beams through the atmo-sphere, Applied physics 10, 129 (1976).

[35] M. Feit, J. Fleck Jr, and A. Steiger, Solution of theschrodinger equation by a spectral method, Journal ofComputational Physics 47, 412 (1982).

[36] R. Babbush, N. Wiebe, J. McClean, J. McClain,H. Neven, and G. K.-L. Chan, Low-depth quantum simu-lation of materials, Physical Review X 8, 011044 (2018).

[37] D. Dong and I. R. Petersen, Quantum control theory andapplications: a survey, IET Control Theory & Applica-tions 4, 2651 (2010).

[38] M. James, Quantum control theory (2014),arXiv:1406.5260 [quant-ph].

[39] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen,and S. J. Glaser, Optimal control of coupled spin dy-namics: design of nmr pulse sequences by gradient as-cent algorithms, Journal of magnetic resonance 172, 296(2005).

[40] F. Motzoi, J. M. Gambetta, S. Merkel, and F. Wilhelm,Optimal control methods for rapidly time-varying hamil-tonians, Physical Review A 84, 022307 (2011).

[41] D. d’Alessandro, Introduction to quantum control and dy-namics (CRC press, 2007).

[42] S. A. Rice, Interfering for the good of a chemical reaction,Nature 409, 422 (2001).

[43] M. Shapiro and P. Brumer, Quantum control of boundand continuum state dynamics, Physics Reports 425, 195(2006).

[44] N. Khaneja, T. Reiss, B. Luy, and S. J. Glaser, Optimalcontrol of spin dynamics in the presence of relaxation,Journal of Magnetic Resonance 162, 311 (2003).

[45] R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang,M. H. Devoret, and R. J. Schoelkopf, Implementing a uni-versal gate set on a logical qubit encoded in an oscillator,Nature Communications 8, 94 (2017).

[46] H. R. Grimsley, S. E. Economou, E. Barnes, and N. J.Mayhall, Adapt-vqe: An exact variational algorithm forfermionic simulations on a quantum computer, arXivpreprint arXiv:1812.11173 (2018).

[47] H. L. Tang, E. Barnes, H. R. Grimsley, N. J. Mayhall,and S. E. Economou, qubit-adapt-vqe: An adaptive al-gorithm for constructing hardware-efficient ansatze ona quantum processor, arXiv preprint arXiv:1911.10205(2019).

[48] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme,Quantum algorithms for fermionic simulations, PhysicalReview A 64, 022319 (2001).

[49] G. Ortiz, J. Gubernatis, E. Knill, and R. Laflamme, Er-ratum: Quantum algorithms for fermionic simulations[phys. rev. a 64, 022319 2001)], Physical Review A 65,029902 (2002).

[50] J. Hubbard, Electron correlations in narrow energybands, Proceedings of the Royal Society of London. SeriesA. Mathematical and Physical Sciences 276, 238 (1963).

[51] S. Yamada, T. Imamura, and M. Machida, 16.447tflops and 159-billion-dimensional exact-diagonalizationfor trapped fermion-hubbard model on the earth sim-ulator, in SC’05: Proceedings of the 2005 ACM/IEEEConference on Supercomputing (IEEE, 2005) pp. 44–44.

[52] K. Masuda and D. Yamamoto, Variational cluster ap-proach to s-wave pairing in heavy-fermion superconduc-tors, Physical Review B 91, 104508 (2015).

[53] M. Guillot, Competition entre l’antiferromagnetisme etla supraconductivite dans le modele et Hubbard appliqueaux cuprates (Universite de Sherbrooke., 2007).

[54] J. Kaczmarczyk, J. Spa lek, T. Schickling, andJ. Bunemann, Superconductivity in the two-dimensionalhubbard model: Gutzwiller wave function solution, Phys-ical Review B 88, 115127 (2013).

[55] P.-L. Dallaire-Demers, M. Stechly, J. F. Gonthier, N. T.Bashige, J. Romero, and Y. Cao, An application bench-mark for fermionic quantum simulations, arXiv preprintarXiv:2003.01862 (2020).

[56] M. Wilson, S. Stromswold, F. Wudarski, S. Had-field, N. M. Tubman, and E. Rieffel, Optimizingquantum heuristics with meta-learning, arXiv preprintarXiv:1908.03185 (2019).

[57] J.-M. Reiner, F. Wilhelm-Mauch, G. Schon, andM. Marthaler, Finding the ground state of the hubbardmodel by variational methods on a quantum computerwith gate errors, Quantum Science and Technology 4,035005 (2019).

[58] G. Verdon, M. Broughton, J. R. McClean, K. J. Sung,R. Babbush, Z. Jiang, H. Neven, and M. Mohseni, Learn-ing to learn with quantum neural networks via clas-sical neural networks, arXiv preprint arXiv:1907.05415(2019).

[59] I. O. Sokolov, P. K. Barkoutsos, P. J. Ollitrault,D. Greenberg, J. Rice, M. Pistoia, and I. Taver-

12

nelli, Quantum orbital-optimized unitary coupled clustermethods in the strongly correlated regime: Can quan-tum algorithms outperform their classical equivalents?,The Journal of Chemical Physics 152, 124107 (2020).

[60] L. Xu, J. T. Lee, and J. Freericks, Test of the uni-tary coupled-cluster variational quantum eigensolver fora simple strongly correlated condensed-matter system,arXiv preprint arXiv:2001.06957 (2020).

[61] S. Hadfield, Z. Wang, B. OGorman, E. G. Rieffel, D. Ven-turelli, and R. Biswas, From the quantum approximateoptimization algorithm to a quantum alternating opera-tor ansatz, Algorithms 12, 34 (2019).

[62] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Sim-ulation of electronic structure hamiltonians using quan-tum computers, Molecular Physics 109, 735 (2011).

[63] X. Bonet-Monroig, R. Sagastizabal, M. Singh, andT. O’Brien, Low-cost error mitigation by symmetry ver-ification, Physical Review A 98, 062339 (2018).

[64] R. Sagastizabal, X. Bonet-Monroig, M. Singh, M. A. Rol,C. Bultink, X. Fu, C. Price, V. Ostroukh, N. Muthusub-ramanian, A. Bruno, et al., Experimental error mitiga-tion via symmetry verification in a variational quantumeigensolver, Physical Review A 100, 010302 (2019).

[65] K. Temme, S. Bravyi, and J. M. Gambetta, Error miti-gation for short-depth quantum circuits, Physical reviewletters 119, 180509 (2017).

[66] S. Endo, S. C. Benjamin, and Y. Li, Practical quan-tum error mitigation for near-future applications, Physi-cal Review X 8, 031027 (2018).

[67] A. Kandala, K. Temme, A. D. Corcoles, A. Mezzacapo,J. M. Chow, and J. M. Gambetta, Extending the compu-tational reach of a noisy superconducting quantum pro-cessor, arXiv preprint arXiv:1805.04492 (2018).

[68] Contributors, Qiskit: An open-source framework forquantum computing (2019).

[69] P. J. OMalley, R. Babbush, I. D. Kivlichan, J. Romero,J. R. McClean, R. Barends, J. Kelly, P. Roushan,A. Tranter, N. Ding, et al., Scalable quantum simula-tion of molecular energies, Physical Review X 6, 031007(2016).

[70] N. Vogt, S. Zanker, J.-M. Reiner, T. Eckl, A. Mar-usczyk, and M. Marthaler, Preparing symmetry bro-ken ground states with variational quantum algorithms,arXiv preprint arXiv:2007.01582 (2020).

[71] M. J. Powell, A direct search optimization method thatmodels the objective and constraint functions by linearinterpolation, in Advances in optimization and numericalanalysis (Springer, 1994) pp. 51–67.

[72] M. J. Powell, Direct search algorithms for optimizationcalculations, Acta numerica 7, 287 (1998).

[73] M. J. Powell, A view of algorithms for optimization with-out derivatives, Mathematics Today-Bulletin of the Insti-tute of Mathematics and its Applications 43, 170 (2007).

Appendix A: Jordan-Wigner fermionic encoding

In the Jordan-Wigner transformation, each fermionicsite is encoded into the state of two qubits with themapping (0, ↑, ↓, ↑↓) 7→ (00, 01, 10, 11). Moreover, the

fermionic ladder operators take the form

ap 7→ σp⊗l<p

Zl,

a†p 7→ σ†p⊗l<p

Zl, (A1)

where σ = |0〉〈1|, Z is the Pauli-Z operator and theindices denote the spin orbitals or qubits. For a lat-tice of L sites, we arrange the N = 2L spin orbitalsas |f1↑ . . . fL↑; f1↓ . . . fL↓〉 with fp ∈ 0, 1 the occupa-tion of spin-orbital p.

With this mapping, hopping terms between spin-orbitals p and q with p < q transform as

a†paq + a†qap 7→1

2(XpXq + YpYq)

q−1⊗l=p+1

Zl, (A2)

where X, Y and Z are Pauli matrices. The productof Z operators, referred to as the JW string, vanisheswhen q = p+ 1. Moreover, the number operator on spin-orbital p, and therefore the onsite Coulomb interactionbetween spin-orbitals p and q take the form

np = a†pap 7→1

2(I − Zp),

npnq 7→1

4(I − Zp − Zq + ZpZq). (A3)

At half-filling, the single Zs coming from the onsite inter-action terms are canceled by similar terms arising fromthe chemical potential, leading to a simple expression forthe potential

V 7→ U

4

L∑i=1

Zi↑Zi↓, (A4)

which is diagonal in the computational basis.

Appendix B: Parametrization of the ansatze

1. Full parametrization

This strategy corresponds to taking all (or almost all)gate angles as variational parameters. This gives the clas-sical optimizer enough freedom to explore the Hilbertspace spanned by the ansatz at the cost of a longer op-timization time. We note that the HEA has, by de-fault, a fully parametrized configuration since all single-qubit gates are parametrized. Moreover, the same strat-egy for VHA consists of assigning one parameter to ev-

ery a†iσajσ + h.c. hopping terms and duplicating the pa-rameter to take into account the two spin orientations.This is because at half-filling and zero total spin, thereis a spin-inversion symmetry which removes the need totreat spins up and down differently. Additionally, every

13

term of the on-site interaction is associated with a vari-ational parameter. The asymptotic scaling of numberof variational parameters for all ansatze is summarizedin Table III for both parametrization strategies.

2. Scalable parametrization

In a scalable parametrization strategy, we employ anumber of variational parameters that is independent ofthe system size. Because there are fewer parameters, weexpect the optimization to be faster, but larger circuitdepths might be necessary to achieve the same accuracyas full parametrization.

Although it is less clear how one would achieve a scal-able parametrization for hardware-efficient approaches,a simple strategy exists for physics-inspired ansatze suchas QOCA. It consists in grouping the individual terms ofthe Hamiltonian into a constant number of sets contain-ing commuting terms. For example, a common way ofgrouping the different terms of the FHM on a 2D latticeis

HFHM = Hh,even+Hh,odd+Hv,even+Hv,odd+HU , (B1)

where the first four terms now group the even and odd,vertical and horizontal hopping terms, while HU collectsthe on-site interaction terms. Note that for the 3D FHM,two additional sets of hopping terms covering the thirddimension would be necessary.

Fullparametrization

Scalableparametrization

HEA 2Ld –VHA (η + 1)Ld (2η + 1)dFT-VHA (η + 1)Ld (η + 1)dQOCA (η + 3)Ld (2η + 3)dsQOCA 3Ld 3d

TABLE III. Asymptotic scaling of the number of variationalparameters of the ansatze of this work for the full and scalableparametrization strategies. These numbers are for periodic η-dimensional Fermi-Hubbard systems of L lattice sites. d is thenumber of layers of the ansatze.

Appendix C: Numerical simulations

All simulations are done using Qiskit Aqua’s VQAtools [68]. Because noise is not considered, a unitarystatevector simulator is used. For simplicity, we also as-sumed all-to-all connectivity of the qubits, although thisis not strictly needed. We chose the COBYLA [71–73]method as the classical optimizer with a maximum num-ber of function evaluation of ∼ 105. This number wasjustified as being reasonable in [21] using experimentallyrealistic arguments.

Whenever possible, we initialize all variational para-meters to zero. With this choice, Hamiltonian-basedansatze implement the identity operator at the start ofthe optimization routine and the variational search be-gins from the initial state. In contrast to a random ini-tialization of the parameters, this strategy also avoidsthe need of doing repeated VQA runs and post-selectingthe best results. However, in the case of short-QOCA,this strategy results in premature convergence of the op-timizer into states close to the initial guess, forcing usto use a random initialization of the parameters. Inter-estingly, even without post-selection, this did not hinderthe convergence capability thanks to the robustness ofQOCA regarding initial conditions.

Finally, all layers of the ansatze are optimized si-multaneously. Further improvement can potentially beachieved by adopting a layer-by-layer optimization strat-egy as in Ref. [23].

For the simulation of the water molecule, we use thePySCF driver to obtain the Hamiltonian as provided

Appendix D: Initial states

In most quantum simulations of the FHM reported inthe literature [21–23, 25, 57, 58, 60], the initial state isthe ground state of the non-interacting FHM i.e. fix-ing U = µ = 0 in Eq. (11). Because the resulting Hamil-tonian is diagonal in Fourier space, this is a convenientchoice because the ground state is readily computed clas-sically. However, preparing this on a quantum computergenerally requires very long quantum circuits as it in-volves the fermionic Fourier transformation. Current im-plementations of this transformation [31, 33, 36] are de-fined only for periodic systems, which limits this initialstate’s applicability. To the best of our knowledge, no im-plementation of an open-boundary-conditions fermionicFourier transformation has been developed to date. Fur-thermore, the ground state of the non-interacting FHMcan be degenerate which makes it difficult to choosewhich one or superposition thereof to use. This chal-lenge is often pointed out as an open problem [21, 60],since in most VQA realization, prior knowledge of thetarget state is used to find the initial state that maxi-mizes the fidelity. It is unclear how one would make thischoice as systems grow computationally intractable.

1. The non-interacting Fermi-Hubbard model

To see how this degeneracy arises, we consider the 1Dnon-interacting FHM (U = µ = 0) with L sites and pe-riodic boundary conditions. In momentum space, theHamiltonian is given by a collection of free fermionicmodes

T = FT T FT† =∑

k,σ=↑,↓

εk c†kσ ckσ, (D1)

14

where the energy spectrum is

εk = −2t cos

(2πk

L

). (D2)

In the above Hamiltonian, c†kσ and ckσ are respectivelythe creation and annihilation fermionic operators of mo-mentum k and spin σ. They are obtained from the real-

space ladder operators a†kσ and akσ and the fermionicFourier transformation as

c†kσ = FT a†kσ FT† =1√L

L−1∑j=0

e−i2πkL j a†jσ, (D3)

ckσ = FT akσ FT† =1√L

L−1∑j=0

ei2πkL j ajσ. (D4)

Because k can only take discrete values, one noticesthat a degeneracy appears when there are energy levelsat εk = 0 since these levels could be occupied or emptywithout affecting the ground state energy. It is straight-forward to see from Eq. (D2) that this can happen onlywhen L = 4l, with l an integer. In this case, there are twovalues of k (corresponding to k = l and k = 3l) whichleads to εk = 0. The degeneracy is therefore 42 = 16since each momentum mode can be empty, occupied bya ↑ or ↓ spin, or both. In the half-filled symmetry sec-tor, the degeneracy is reduced to ( 4

2 ) = 6. Note that inthe case L 6= 4l, the ground state of the non-interactingFHM is not degenerate and is a simple basis state inmomentum space.

As mentioned above, this occasional degeneracy makesit difficult to guess which basis state (or superpositionthereof) is the best initial state to use in a VQA. Al-though, one can select states that respect certain desiredproperties such as particle number, total spin and totalmomentum.

Typically, the degeneracy at L = 4l can be lifted byapplying a small perturbative Coulomb interaction U . Inthis case, the ground state of the non-interacting FHM

becomes a superposition of basis states in Fourier space.One must apply the FT† in order to transform this initialstate into real space for the VQA.

2. Choosing and preparing the initial states

In the case of L = 4 (or 2 × 2), we computed thefidelity of the 16 degenerate ground states of Eq. (D1)with respect to the target ground state and post-selectedthe ones leading to the highest fidelity. This strategy is,of course, not scalable and therefore it remains unclearhow one would proceed in practice in the case where thefidelity with the target ground state cannot be computedbeforehand.

In the present case, this strategy yields two groundstates with a fidelity of ≈ 0.425 with respect to theground state of the full model. Labeling the spin or-bitals |f1↑ . . . fL↑; f1↓ . . . fL↓〉, these two states in realspace are

|Ω(1)T 〉 = FT† |1100 ; 1100〉, (D5)

|Ω(2)T 〉 = FT† |1001 ; 1001〉. (D6)

Preparing these two states requires applying Pauli-Xgates on selected qubits followed by the fermionic Fouriertransformation, something which requires long quantumcircuits [31, 33, 36].

Adding a small perturbation U = 1 × 10−5t, we find

that the following superposition of |Ω(1)T 〉 and |Ω(2)

T 〉 yieldsa significantly larger fidelity to the true ground state of ≈0.85:

|ΩT 〉 =|Ω(1)T 〉 − |Ω

(2)T 〉√

2

= FT†|1100 ; 1100〉 − |1001 ; 1001〉√

2. (D7)

This, however, increases the complexity of the initialstate preparation.